Introduction - Georgia Institute of Technologypeople.math.gatech.edu/~mbaker/pdf/hyperelliptic.pdf[Bak07].) We then prove that the graph-theoretic analogues of conditions (H1)-(H5)

HARMONIC MORPHISMS AND HYPERELLIPTIC GRAPHS

MATTHEW BAKER AND SERGUEI NORINE

Abstract. We study harmonic morphisms of graphs as a natural discrete analogue of holomorphicmaps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by aharmonic morphism, and present a discrete analogue of the canonical map from a Riemann surfaceto projective space. We also discuss several equivalent formulations of the notion of a hyperellipticgraph, all motivated by the classical theory of Riemann surfaces. As an application of our results,we show that for a 2-edge-connected graph G which is not a cycle, there is at most one involutionι on G for which the quotient G/ι is a tree. We also show that the number of spanning trees ina graph G is even if and only if G admits a non-constant harmonic morphism to the graph B2

consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula andour results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.

1. Introduction

1.1. Notation and terminology. Throughout this paper, a Riemann surface will mean a com-pact, connected one-dimensional complex manifold, and (unless otherwise specified) a graph willmean a finite, connected multigraph without loop edges. A graph with no multiple edges is calledsimple. We will denote by V (G) and E(G), respectively, the set of vertices and edges of G. For avertex x ∈ V (G) and an edge e ∈ E(G), we write x ∈ e if e is incident to x.

We denote by g(G) := |E(G)| − |V (G)| + 1 the genus of G; this is the dimension of the cyclespace of G. (In graph theory, the term “genus” is traditionally used for a different concept, namely,the smallest genus of any surface in which the graph can be embedded, and the integer g = g(G)is called the “cyclomatic number” of G. We call g the genus of G in order to highlight the analogywith Riemann surfaces.)

For k ≥ 2, a graph G is called k-edge-connected if G−W is connected for every set W of at mostk − 1 edges of G. (By convention, we consider the trivial graph having one vertex and no edges tobe k-edge-connected for all k.) Alternatively, define a cut to be the set of all edges connecting avertex in V1 to a vertex in V2 for some partition of V (G) into disjoint subsets V1 and V2. Then Gis k-edge-connected if and only if every non-empty cut has size at least k.

A bridge is an edge of G whose deletion increases the number of connected components of G. A(connected) graph is 2-edge-connected if and only if it contains no bridge.

Finally, if A ⊆ V (G), we denote by χA : V (G) → 0, 1 the characteristic function of A, and forx ∈ A we let outdegA(x) denote the number of edges e = xy in E(G) with y 6∈ A.

1.2. Motivation and discussion of main results. In [BN07], the authors investigated somenew analogies between graphs and Riemann surfaces, formulating the notion of a linear systemon a graph and proving a graph-theoretic analogue of the classical Riemann-Roch theorem. Thetheory of linear systems on graphs has applications to understanding the Jacobian of a finite graph,a group which is analogous to the Jacobian of a Riemann surface, and which has appeared in many

Date: November 17, 2008.The second author was partially supported by NSF under Grant No. DMS-0200595 and DMS-0701033.

1

2 MATTHEW BAKER AND SERGUEI NORINE

different guises throughout the literature (e.g., as the “Picard group” in [BdlHN97], the “criticalgroup” in [Big97], the “sandpile group” in [Dha90], and the “group of components” in [Lor91]).

The present paper can be viewed as a natural sequel to [BN07]. In classical algebraic geometry,one is usually interested not just in Riemann surfaces themselves, but also in the holomorphic mapsbetween them. Thus, we are naturally led to ask: what is the “correct” graph-theoretic analogueof a holomorphic map between Riemann surfaces? In other words, is there a category consisting ofgraphs, together with certain maps between them, which closely mirrors the category of Riemannsurfaces, together with the holomorphic maps between them? In this paper, we hope to convincethe reader that the notion of a harmonic morphism of graphs, introduced by Urakawa in [Ura00],has essentially all of the desired features.

Actually, since we want to allow graphs with multiple edges, we need to slightly modify thedefinition of a harmonic morphism from [Ura00], since Urakawa assumes that all of his graphs aresimple. Recall that a holomorphic map φ : X → X ′ between Riemann surfaces is one which locallypulls back holomorphic functions on X ′ to holomorphic functions on X. Although the notion ofa holomorphic function does not make sense in the context of graphs, there is a natural notion ofa harmonic function (see (1.7) below for the definition). Urakawa defines a harmonic morphismφ : V (G) → V (G′) between simple graphs G,G′ to be a function which locally pulls back harmonicfunctions on G′ to harmonic functions on G, i.e., a function such that for every x ∈ V (G) and everyfunction f : V (G′) → R which is harmonic at φ(x), the composition f φ is harmonic at x. Whatmakes this a workable and useful notion is Theorem 2.5 from [Ura00], which asserts that a functionφ : V (G) → V (G′) between simple graphs is a harmonic morphism if and only if φ is horizontallyconformal, meaning that:

(HC1) For all adjacent vertices x, y ∈ V (G), we have either φ(x) = φ(y) or φ(x) is adjacent toφ(y), and

(HC2) For all x ∈ V (G), the quantity

|y ∈ V (G) | φ(y) = y′ and y is adjacent to x|

is the same for all y′ ∈ V (G′) adjacent to x′ := φ(x).

However, the equivalence between harmonic morphisms and horizontally conformal maps failsfor graphs which are not simple (c.f. Remark 2.10 below). Because of this, we take an analogue of(HC1) and (HC2) as our definition of a harmonic morphism between multigraphs (see Definition 2.1for a precise definition). A harmonic morphism in this sense does indeed locally pull back harmonicfunctions to harmonic functions, but the converse does not always hold.

One of the key features of defining harmonic morphisms in terms of horizontal conformality isthat given a harmonic morphism φ : G → G′, it is possible to assign a well-defined multiplicitymφ(x) to each vertex x ∈ V (G) (analogous to the ramification index eφ(x) at x ∈ X for a non-constant holomorphic map φ : X → X ′ between Riemann surfaces) in such a way that the sumdeg(φ) of the multiplicities at all vertices mapping to a given vertex x′ ∈ V (G′) is independent ofx′. We define the degree of φ to be this number.

Harmonic morphisms between graphs enjoy numerous properties analogous to classical propertiesfrom algebraic geometry. For example, if φ : X → X ′ is a non-constant holomorphic map of degreedeg(φ) between Riemann surfaces having genus g and g′, respectively, then:

(RS1) φ is surjective and g ≥ g′.(RS2) The Riemann-Hurwitz formula 2g − 2 = deg(φ)(2g′ − 2) +

∑x∈X (eφ(x)− 1) holds.

HARMONIC MORPHISMS AND HYPERELLIPTIC GRAPHS 3

(RS3) φ induces functorial maps φ∗ : Jac(X) → Jac(X ′) and φ∗ : Jac(X ′) → Jac(X) between theJacobians of X and X ′.

(RS4) φ induces functorial maps φ∗ : Ω1(X) → Ω1(X ′) and φ∗ : Ω1(X ′) → Ω1(X) between thecomplex vector spaces Ω1(X) and Ω1(X ′) of holomorphic 1-forms on X and X ′, respectively.

(RS5) If D is a divisor on X, then dim |φ∗(D)| ≥ dim |D|, where |D| denotes the complete linearsystem associated to D. In particular, if X is hyperelliptic and g(X ′) ≥ 2, then X ′ ishyperelliptic as well.

We will prove graph-theoretic analogues of all of these classical facts. We will also describe somesituations in which the naive analogue of certain classical facts does not hold. For example, inalgebraic geometry the map φ∗ : Jac(X ′) → Jac(X) is sometimes injective and sometimes not;more precisely, it is known that φ∗ fails to be injective if and only if φ has a nontrivial unramifiedabelian subcover. However, the analogous map φ∗ : Jac(G′) → Jac(G) in the graph-theoreticcontext turns out to always be injective; this appears to be a rather subtle fact with some usefulapplications.

As a basic testing ground for our “dictionary” between graphs and Riemann surfaces, we considerin detail the graph-theoretic analogue of a hyperelliptic Riemann surface. This is particularlyinteresting because classically, there are many different equivalent characterizations of what itmeans for a Riemann surface X of genus at least 2 to be hyperelliptic. As just a sample, wemention the following:

(H1) There exists a divisor D of degree 2 on X for which r(D) := dim |D| is equal to 1.(H2) There exists an involution ι for which X/ι has genus 0.(H3) There is a degree 2 holomorphic map φ : X → P1.(H4) There is an automorphism ι of X for which ι∗ : Jac(X) → Jac(X) is multiplication by −1.(H5) There is an automorphism ι of X for which ι∗ : Ω1(X) → Ω1(X) is multiplication by −1.(H6) The symmetric square S(2)

x0 : Div2+(X) → Jac(X) of the Abel-Jacobi map (relative to some

base point x0 ∈ X) is not injective.(H7) The canonical map ψX : X → P(Ω1(X)) is not injective.When any of these equivalent conditions are satisfied, there is a unique automorphism ι satisfying

(H2), (H4), and (H5), called the hyperelliptic involution.

For a 2-edge-connected graph G of genus at least 2, we take the analogue of (H1) to be thedefinition of what it means for G to be hyperelliptic. (This definition was already introduced in[Bak07].) We then prove that the graph-theoretic analogues of conditions (H1)-(H5) above are allequivalent to one another, and that the hyperelliptic involution ι on a graph satisfying any of theseconditions is unique. However, in the graph-theoretic context it turns out that (H1) ⇒ (H6) ⇔(H7), so that hyperelliptic graphs satisfy the analogues of conditions (H6) and (H7), but there arenon-hyperelliptic 2-edge-connected graphs G of genus at least 2 which also satisfy these conditions.In fact, we will see that the graph-theoretic analogues of conditions (H6) and (H7) are equivalentto the condition that G is not 3-edge-connected.

As an application of our results, and to illustrate another difference with the theory of Riemannsurfaces, we conclude our paper with a discussion of Weierstrass points on hyperelliptic graphs.(The notion of a Weierstrass points on graphs was introduced in [Bak07]; see §5 for a definition.)Classically, a hyperelliptic Riemann surface of genus g ≥ 2 possesses exactly 2g + 2 Weierstrasspoints, namely, the fixed points of the hyperelliptic involution, and every Riemann surface of genusat least 2 has Weierstrass points. The situation for graphs is less orderly, as there are infinitefamilies of graphs having no Weierstrass points at all. Using our rather precise knowledge about


the structure of hyperelliptic graphs, we give a classification of all hyperelliptic graphs having noWeierstrass points.

Occasionally, our foundational results on harmonic morphisms and hyperelliptic graphs lead toapplications to more traditional-sounding graph-theoretic questions. For example, as a consequenceof our study of hyperelliptic graphs, we will show that for a 2-edge-connected graph G of genus atleast 2, there is at most one involution ι on G whose quotient is a tree. We also show that thenumber κG of spanning trees in a graph G is even if and only if G admits a non-constant degree 2harmonic morphism to the graph B2 consisting of 2 vertices connected by 2 edges.

Although in this paper we view our graph-theoretic results as “analogous” to classical resultsfrom algebraic geometry, there is in fact a closer connection between the two worlds than one mightat first imagine. One such connection arises from the specialization of divisors on an arithmeticsurface, and is explored in [Bak07]. We expect that the ideas in the present paper will help spurfurther interactions between graph theory, on the one hand, and arithmetic, algebraic, and tropicalgeometry on the other.

It would be interesting to prove analogues of the results in the present paper for metric graphs,and more generally for tropical curves, but we have not attempted to do so here. It would alsobe interesting to generalize some of our results to higher dimensions. At least in the context ofRiemannian polyhedra (which are higher-dimensional analogues of metric graphs), there is alreadya rich literature concerning the notion of a harmonic morphism (see, e.g., [EF01]). However, itappears that the questions being addressed in [EF01] and the references therein are of a somewhatdifferent flavor than the ones which we study here.

We have endeavored to make this paper as self-contained as possible. Therefore, we summarizein §1.3 below all of the facts from [BN07] which we will be using. We have also rewritten certainproofs from [Ura00], because our notation differs somewhat from Urakawa’s, and because we workin the somewhat more general setting of multigraphs. A good reference for many of the facts aboutRiemann surfaces which we will be discussing in this paper is [Mir95], and a basic graph theoryreference is [Bol98].

1.3. Background material from [BN07]. Following [BN07], we denote by Div(G) the free abeliangroup on V (G). We refer to elements of Div(G) as divisors on G. We can write each elementD ∈ Div(G) uniquely as D =

∑x∈V (G)D(x)(x) with D(x) ∈ Z. We say that D is effective, and

write D ≥ 0, if D(x) ≥ 0 for all x ∈ V (G). For D ∈ Div(G), we define the degree of D by theformula deg(D) =

∑x∈V (G)D(x). We denote by Div0(G) the subgroup of Div(G) consisting of

divisors of degree zero. Finally, we denote by Divk+(G) = E ∈ Div(G) : E ≥ 0, deg(E) = k the

set of effective divisors of degree k on G.Let C0(G,Z) be the group of Z-valued functions on V (G). For f ∈ C0(G,Z), we define the

divisor of f by the formula

div(f) =∑

x∈V (G)

∑e=xy∈E(G)

(f(x)− f(y)) (x).

The divisor of f can be naturally identified with the graph-theoretic Laplacian of f . Divisorsof the form div(f) for some f ∈ C0(G,Z) are called principal; we denote by Prin(G) the groupof principal divisors on G. It is easy to see that every principal divisor has degree zero, so thatPrin(G) is a subgroup of Div0(G).

The Jacobian of G, denoted Jac(G), is defined to be the quotient group

Jac(G) = Div0(G)/Prin(G).


One can show using Kirchhoff’s Matrix-Tree Theorem (c.f. [Big97, §14]) that Jac(G) is a finiteabelian group of order κG, where κG is the number of spanning trees in G.

We define an equivalence relation ∼G on Div(G) by writing D ∼G D′ if and only if D − D′ ∈Prin(G), and set

|D| = E ∈ Div(G) : E ≥ 0 and E ∼G D.We refer to |D| as the (complete) linear system associated to D, and when D ∼ D′ we call thedivisors D and D′ linearly equivalent. We will usually just write D ∼ D′, rather than D ∼G D′,when the graph G is understood.

For later use, we note the following simple fact about the linear equivalence relation on G:

Lemma 1.1. We have (x) ∼ (y) for all x, y ∈ V (G) if and only if G is a tree.

Proof. This follows from the fact that | Jac(G)| = κG, together with the observation that the groupDiv0(G) is generated by the divisors of the form (x)− (y) with x, y ∈ V (G).

Given a divisor D on G, define r(D) = −1 if |D| = ∅, and otherwise set

r(D) = maxk ∈ Z : |D − E| 6= ∅ ∀ E ∈ Divk+(G).

Note that r(D) depends only on the linear equivalence class of D, and therefore is an invariantof the linear system |D|. When we wish to emphasize the underlying graph G, we will sometimeswrite rG(D) instead of r(D).

For later use, we recall from [BN07, Lemma 2.1] the following simple lemma:

Lemma 1.2. For all D,D′ ∈ Div(G) such that r(D), r(D′) ≥ 0, we have r(D+D′) ≥ r(D)+r(D′).

We define the canonical divisor on G to be

KG =∑

x∈V (G)

(deg(x)− 2)(x).

We have deg(KG) = 2g−2, where g = |E(G)|−|V (G)|+1 is the genus of G (or, in more traditionallanguage, cyclomatic number of G).

The following result is proved in [BN07, Theorem 1.12]:

Theorem 1.3 (Riemann-Roch for graphs). Let D be a divisor on a graph G. Then

r(D)− r(KG −D) = deg(D) + 1− g.

As a consequence of Lemma 1.2 and Theorem 1.3, we have the following graph-theoretic analogueof a classical result known as Clifford’s theorem (see [BN07, Corollary 3.5] for a proof):

Corollary 1.4 (Clifford’s Theorem for graphs). Let D be a divisor on a graph G for which |D| 6= ∅and |KG −D| 6= ∅. Then

r(D) ≤ 12

deg(D) .

The next result (Theorem 3.3 from [BN07]) is very useful for computing r(D) in specific examples,and also plays an important role in the proof of Theorem 1.3. For each linear ordering < on V (G),we define a corresponding divisor ν ∈ Div(G) of degree g − 1 by the formula

ν =∑

x∈V (G)

(|e = xy ∈ E(G) : y < x| − 1)(x).

Theorem 1.5. For every D ∈ Div(G), exactly one of the following holds:


(1) r(D) ≥ 0; or(2) r(ν −D) ≥ 0 for some divisor ν associated to a linear ordering < of V (G).

Finally, we recall some facts from [BN07] and [BdlHN97] about the graph-theoretic analogue ofthe Abel-Jacobi map from a Riemann surface to its Jacobian.

For a fixed base point x0 ∈ V (G), we define the Abel-Jacobi map Sx0 : G → Jac(G) by theformula

(1.6) Sx0(x) = [(x)− (x0)] .

The map Sx0 can be characterized by the following universal property (see §3 of [BdlHN97]). Amap ϕ : G→ A from V (G) to an abelian group A is called harmonic if for each x ∈ V (G), we have

(1.7) deg(x) · ϕ(x) =∑

e=xy∈E(G)

ϕ(y) .

Then Sx0 is universal among all harmonic maps from G to abelian groups sending x0 to 0, in thefollowing precise sense:

Lemma 1.8. If ϕ : G → A is a harmonic map sending x0 ∈ V (G) to 0, then there is a uniquegroup homomorphism ψ : Jac(G) → A such that ϕ = ψ Sx0.

We also define, for each integer k ≥ 1, a map S(k)x0 : Divk

+(G) → Jac(G) by

S(k)x0

((x1) + · · ·+ (xk)) = Sx0(x1) + Sx0(x2) + · · ·+ Sx0(xk) .

The following result is proved in [BN07, Theorem 1.8]:

Theorem 1.9. The map S(k)x0 is injective if and only if G is (k + 1)-edge-connected.

2. Harmonic morphisms

2.1. Definition and basic properties of harmonic morphisms. Harmonic morphisms be-tween simple graphs were defined and studied in [Ura00]. Here, we reproduce some definitionsfrom [Ura00], but with several variations due to the fact that we allow our graphs to have multipleedges.

Let G,G′ be graphs. A function φ : V (G) ∪ E(G) → V (G′) ∪ E(G′) is said to be a morphismfrom G to G′ if φ(V (G)) ⊆ V (G′), and for every edge e ∈ E(G) with endpoints x and y, eitherφ(e) ∈ E(G′) and φ(x), φ(y) are the endpoints of φ(e), or φ(e) ∈ V (G′) and φ(e) = φ(x) = φ(y).We write φ : G → G′ for brevity. If φ(E(G)) ⊆ E(G′) then we say that φ is a homomorphism.A bijective homomorphism is called an isomorphism, and an isomorphism φ : G → G is called anautomorphism.

We now come to the key definition in this paper.

Definition. A morphism φ : G → G′ is said to be harmonic (or horizontally conformal) if for allx ∈ V (G), y ∈ V (G′) such that y = φ(x), the quantity |e ∈ E(G)|x ∈ e, φ(e) = e′| is the samefor all edges e′ ∈ E(G′) such that y ∈ e′.

Remark 2.1. One can check directly from the definition that the composition of two harmonic mor-phisms is again harmonic. Therefore the class of all graphs, together with the harmonic morphismsbetween them, forms a category.


Let φ : G→ G′ be a morphism and let x ∈ V (G). Define the vertical multiplicity of φ at x by

vφ(x) = |e ∈ E(G) |φ(e) = φ(x)|.This is simply the number of vertical edges incident to x, where an edge e is called vertical ifφ(e) ∈ V (G) (and is called horizontal otherwise).

If φ is harmonic and |V (G′)| > 1, we define the horizontal multiplicity of φ at x by

mφ(x) = |e ∈ E(G)|x ∈ e, φ(e) = e′|for any edge e′ ∈ E(G) such that φ(x) ∈ e′. By the definition of a harmonic morphism, mφ(x) isindependent of the choice of e′. When |V (G′)| = 1, we define mφ(x) to be 0 for all x ∈ V (G).

If deg(x) denotes the degree of a vertex x, we have the following basic formula relating thehorizontal and vertical multiplicities:

(2.2) deg(x) = deg(φ(x))mφ(x) + vφ(x).

We say that a harmonic morphism φ : G→ G′ is non-degenerate ifmφ(x) ≥ 1 for every x ∈ V (G).(The motivation for this definition comes from Theorem 5.12 below.)

If |V (G′)| > 1, we define the degree of a harmonic morphism φ : G→ G′ by the formula

(2.3) deg(φ) := |e ∈ E(G) | φ(e) = e′|for any edge e′ ∈ E(G′). (When |V (G′)| = 1, we define deg(φ) to be 0.) By the following lemma(c.f. [Ura00, Lemma 2.12]), the right-hand side of (2.3) does not depend on the choice of e′ (andtherefore deg(φ) is well-defined):

Lemma 2.4. The quantity |e ∈ E(G) | φ(e) = e′| is independent of the choice of e′ ∈ E(G′).

Proof. Let y ∈ V (G′), and suppose there are two edges e′, e′′ ∈ E(G′) incident to y. Since φ ishorizontally conformal, for each x ∈ V (G) with φ(x) = y we have

|e ∈ E(G) | x ∈ e, φ(e) = e′| = |e ∈ E(G) | x ∈ e, φ(e) = e′′|.Therefore

(2.5)

|e ∈ E(G) | φ(e) = e′| =∑

x∈φ−1(y)

|e ∈ E(G) | x ∈ e, φ(e) = e′|

=∑

x∈φ−1(y)

|e ∈ E(G) | x ∈ e, φ(e) = e′′|

= |e ∈ E(G) | φ(e) = e′′|.

Now suppose e′, e′′ are arbitrary edges of G′. Since G is connected, the result follows by applying(2.5) to each pair of consecutive edges in any path connecting e′ and e′′.

According to the next result, the degree of a harmonic morphism φ : G→ G′ is just the numberof preimages under φ of any vertex of G′, counting multiplicities:

Lemma 2.6. For any vertex y ∈ G′, we have

deg(φ) =∑

x∈V (G)φ(x)=y

mφ(x).


Proof. Choose an edge e′ ∈ E(G′) with y ∈ e′. Then∑x∈φ−1(y)

mφ(x) =∑

x∈φ−1(y)

∑e∈φ−1(e′), x∈e

1

= |φ−1(e′)| = deg(φ).

As with morphisms of Riemann surfaces in algebraic geometry, a harmonic morphism of graphsmust be either constant or surjective. More generally, one has the following:

Lemma 2.7. Let φ : G→ G′ be a harmonic morphism with |V (G′)| > 1. Then deg(φ) = 0 if andonly if φ is constant, and deg(φ) > 0 if and only if φ is surjective.

Proof. If φ is constant, then clearly deg(φ) = 0. Moreover, it follows from Lemmas 2.4 and 2.6 thatφ is surjective if and only if deg(φ) > 0. So it remains only to show that if deg(φ) = 0, then φ isconstant. For this, suppose we have φ(x) = y. Since mφ(x) = 0, it follows that φ(e) = y for everyedge e with x ∈ e. Thus φ(x′) = y for every neighbor x′ of x. As G is connected, it follows thatevery vertex and every edge of G is mapped under φ to y.

2.2. Harmonic morphisms and harmonic maps to abelian groups. Recall that given agraph G and an abelian group A, a function f : V (G) → A is said to be harmonic at x ∈ V (G) if∑

e=xy∈E(G)

(f(x)− f(y)) = 0.

A morphism φ : G→ G′ is said to be A-harmonic if for any y = φ(x) and any function f : V (G′) →A harmonic at y, the function f φ is harmonic at x.

Proposition 2.8. Let G and G′ be graphs, and let φ : G→ G′ be a harmonic morphism. Then φis A-harmonic for every abelian group A.

Proof. (c.f. Lemma 2.11 of [Ura00]) Let x ∈ V (G), y ∈ V (G′) be such that y = φ(x), and letf : V (G′) → A be harmonic at y, i. e.∑

e=zy∈E(G′)

f(z) = deg(y)f(y).

Then we have

∑e=zx∈E(G)

f(φ(z)) =∑

e=zx∈E(G)φ(e)=y

f(φ(z)) +∑

e′=z′y∈E(G′)

∑e=zx∈E(G)

φ(e)=e′

f(φ(z))

= vφ(x)f(y) +

∑e′=z′y∈E(G′)

mφ(x)f(z′)

= vφ(x)f(y) +mφ(x) deg(y)f(y)

= (vφ(x) +mφ(x) deg(φ(x)))f(φ(x))

= deg(x)f(φ(x)) (by (2.2)),

as desired.

If G′ is a simple graph (i.e., without multiple edges), then the converse of Proposition 2.8 alsoholds:


Proposition 2.9. If G′ is a simple graph, then for a morphism φ : G → G′, the following areequivalent:

(1) φ is harmonic (i.e., horizontally conformal).(2) φ is A-harmonic for every abelian group A.(3) φ is R-harmonic.

Proof. It follows from Proposition 2.8 that (1) implies (2), and it is immediate that (2) implies (3).It remains to show that (3) implies (1), which we do following [Ura00, Lemma 2.7].

For a vertex x ∈ V (G) and an edge e′ ∈ E(G′) such that φ(x) ∈ e′, let

kx(e′) = |e ∈ E(G)|x ∈ e, φ(e) = e′|.We need to prove that kx(e′) is independent of the choice of e′. Let φ(x) = y, let e′ = yz, and definea function fe′ : V (G′) → R as follows. Let fe′(z) = 1, let fe′(y) = 1/deg(y), and let fe′(w) = 0 forw ∈ V (G′)\y, z. Then fe′ is harmonic at y, so by (3), fe′ φ is harmonic at x. It follows that

deg(x)deg(y)

= deg(x)fe′(φ(x)) =∑

e=xw∈E(G)

fe′(φ(w))

=∑

e=xw∈E(G)φ(w)=y

fe′(y) +∑

e=xw∈E(G)φ(w)=z

fe′(z)

=vφ(x)deg(y)

+ kx(e′) (since G′ is simple).

Therefore kx(e′) = (deg(x)− vφ(x))/deg(φ(x)) is independent of the choice of e′, as desired.

Remark 2.10. If G′ is not simple, then the converse of Proposition 2.9 may fail, as one sees easilyby taking G to be the graph with 2 vertices x, y connected by a single edge e, G′ to be the graphwith 2 vertices x′, y′ connected by two edges e′1, e

′2, and φ : G → G′ to be the morphism which

sends x, y to x′, y′, respectively, and e to e′1.

2.3. The Riemann-Hurwitz formula for graphs. Let φ : G → G′ be a harmonic morphism.We define the push-forward homomorphism φ∗ : Div(G) → Div(G′) by

(2.11) φ∗(D) =∑

x∈V (G)

D(x)(φ(x)).

Similarly, we define the pullback homomorphism φ∗ : Div(G′) → Div(G) by

(2.12) φ∗(D′) =∑

y∈V (G′)

∑x∈V (G)φ(x)=y

mφ(x)D′(y)(x).

Lemma 2.13. If φ : G → G′ is a harmonic morphism and D′ ∈ Div(G′), then deg(φ∗(D′)) =deg(φ) · deg(D′).

Proof. This follows from Lemma 2.6 and the definition of φ∗.

A basic fact about harmonic morphisms of graphs is that one has the following analogue of theclassical Riemann-Hurwitz formula from algebraic geometry:

Theorem 2.14 (Riemann-Hurwitz for graphs). Let G,G′ be graphs, and let φ : G → G′ be aharmonic morphism. Then:


(1) The canonical divisors on G and G′ are related by the formula

(2.15) KG = φ∗KG′ +RG,

whereRG = 2

∑x∈V (G)

(mφ(x)− 1)(x) +∑

x∈V (G)

vφ(x)(x).

(2) If G,G′ have genus g and g′, respectively, then

(2.16) 2g − 2 = deg(φ)(2g′ − 2) +∑

x∈V (G)

(2(mφ(x)− 1) + vφ(x)) .

(3) If φ is non-constant, then 2g − 2 ≥ deg(φ)(2g′ − 2) and g ≥ g′.

Proof. By definition, we have (φ∗KG′)(x) = mφ(x)(deg(φ(x))− 2). On the other hand, by (2.2) wehave

KG(x) = deg(x)− 2 = deg(φ(x))mφ(x) + vφ(x)− 2

= (φ∗KG′)(x) + 2mφ(x) + vφ(x)− 2 = (φ∗KG′ +RG)(x)

for every x ∈ V (G), which proves (1). Part (2) follows immediately from Lemma 2.13 uponcomputing the degrees of the divisors on both sides of (2.15). In order to verify (3), we claim that ifφ is non-constant then deg(RG) ≥ 0. This is clear if G has no vertical leaves (i.e., degree 1 verticesx having mφ(x) = 0). On the other hand, suppose x is a vertical leaf, and let e = xy be the uniqueedge with x ∈ e. Then if G is the graph obtained by contracting e to y, the induced map G→ G′ isstill harmonic and non-constant, and deg(RG) = deg(RG). Continuing in this way, we can reduceour claim to the already established case where G has no vertical leaves.

Remark 2.17. In the classical Riemann-Hurwitz formula from algebraic geometry, for a non-constantholomorphic map φ : X → X ′ between Riemann surfaces of genus g and g′, respectively, one has

2g − 2 = deg(φ)(2g′ − 2) +∑x∈X

(eφ(x)− 1) ,

where eφ(x) denotes the ramification index of φ at x. Note that there is no analogue in algebraicgeometry of the “vertical multiplicities” vφ(x), and there is an extra factor of 2 in the right-handside of (2.16). Also, note that for Riemann surfaces one has a linear equivalence KX ∼ φ∗KX′ +RX

(which is all that can be expected, since there are just canonical divisor classes on X and X ′, notcanonical divisors), but in (2.15) we have an actual equality of divisors.

3. Examples

In this section, we give some examples of harmonic and non-harmonic morphisms.

Example 3.1 (A harmonic morphism). The morphism shown in Figure 1 is harmonic, with hori-zontal and vertical multiplicities mφ(x) and vφ(x), respectively, labeled next to the correspondingvertices.

Example 3.2 (Harmonic morphisms to trees). Every graph G admits a non-constant harmonicmorphism to a tree. More precisely, suppose |V (G)| ≥ 2 and let x ∈ V (G) be a vertex of degreek ≥ 1. Let T be the graph consisting of two vertices a, b connected by a single edge e′, and let φbe the morphism sending x to a and every y ∈ V (G)\x to b, and sending an edge e ∈ E(G) toe′ if x ∈ e, and to b otherwise. Then φ is a harmonic morphism of degree k.


m=1

m=2

m=2

m=1

m=0

v=1m=1

m=2

m=1

m=1

m=1

m=0v=1

m=1

m=1

m=1v=1

v=1

G’

G

v=2

v=2

Figure 1. A harmonic morphism φ : G→ G′ of degree 3.

Example 3.3 (Automorphisms). A graph automorphism α : G→ G is a non-degenerate harmonicmorphism of degree 1.

Example 3.4 (Coverings). A homomorphism φ : G→ G′ is a covering of degree d ≥ 1 if deg(x) =deg(φ(x)) for every x ∈ V (G) and φ−1(e′) consists of d disjoint edges for every edge e′ ∈ E(G′).A covering is a harmonic morphism; more precisely, a covering morphism is the same thing as aharmonic morphism for which mφ(x) = 1 and vφ(x) = 0 for all x ∈ V (G).

Example 3.5 (Collapsing). Let p ∈ V (G) be a cut vertex, so that G can be partitioned into twosubsets G1 and G2 which intersect only at p. We define the collapsing of G relative to G1 to bethe graph G′ obtained by contracting all vertices and edges in G1 to p. Let φ : G → G′ be themorphism which sends G1 to p and is the identity on G2. Then if |V (G2)| > 1, it is easy to seethat φ is a harmonic morphism of degree 1 (c.f. [Ura00, Proposition 4.2]).

Example 3.6 (Contracting bridges is not harmonic). Let e ∈ E(G) be a bridge, and let G be thegraph obtained by contracting e. Then there is an evident contraction morphism ρ : G→ G whichis surjective on both vertices and edges. However, ρ is not in general a harmonic morphism, as inFigure 2.

G

G’

Figure 2. A non-harmonic morphism ρ : G→ G′.


4. Functorial properties

In this section, we discuss how harmonic morphisms between graphs induce different kinds offunctorial maps between divisor groups, Jacobians, and harmonic 1-forms.

4.1. Induced maps on Jacobians. In §2.3, we introduced homomorphisms φ∗ : Div(G) →Div(G′) and φ∗ : Div(G′) → Div(G) associated to a harmonic morphism φ : G → G′. Thesehomomorphisms are related by the following simple formula:

Lemma 4.1. Let φ : G→ G′ be a harmonic morphism, and let D′ ∈ Div(G′). Then φ∗(φ∗(D′)) =deg(φ)D′.

Proof. This follows from Lemma 2.6 and the definitions of φ∗ and φ∗.

Suppose φ : G → G′ is a harmonic morphism and that f : V (G) → A and f ′ : V (G′) → A arefunctions, where A is an abelian group. We define φ∗f : V (G′) → A by

φ∗f(y) :=∑

x∈V (G)φ(x)=y

mφ(x)f(x)

and φ∗f ′ : V (G) → A byφ∗f ′ := f ′ φ.

Proposition 4.2. Let φ : G→ G′ be a harmonic morphism, let f : V (G) → Z and f ′ ∈ V (G′) → Z.Then

(4.3) φ∗(div(f)) = div(φ∗f)

and

(4.4) φ∗(div(f ′)) = div(φ∗f ′).

Proof. We start by proving (4.3). We have

div(f) =∑

e=xy∈E(G)

(f(x)− f(y))((x)− (y)).

By the linearity of φ∗, we have

(4.5) φ∗(div(f)) =∑

e=xy∈E(G)

(f(x)− f(y))(φ(x)− φ(y)).

By the definition of φ∗f , we have(4.6)

div(φ∗f) =∑

e′=x′y′∈E(G′)

∑x∈V (G), φ(x)=x′

mφ(x)f(x)−∑

y∈V (G), φ(y)=y′

mφ(y)f(y)

((x′)− (y′)

).

Note that terms in (4.5) corresponding to edges in φ−1(V (G′)) are zero. Therefore, to derive (4.3)from (4.5) and (4.6), it suffices to prove that∑

e=xy∈φ−1(e′)

(f(x)− f(y)) =∑

x∈V (G), φ(x)=x′

mφ(x)f(x)−∑

y∈V (G), φ(y)=y′

mφ(y)f(y)

for every edge e′ = x′y′ ∈ E(G′). This last identity holds by the definition of mφ.


We now prove (4.4). Let D′ := div(f ′). We have D′(y) = deg(y)f ′(y) −∑

e=zy∈E(G′) f′(z) for

every y ∈ V (G′), so by the definition of φ∗, we have

(4.7) (φ∗D′)(x) = mφ(x)D′(φ(x)) = mφ(x) deg(φ(x))f ′(φ(x))−mφ(x)∑

e=zφ(x)∈E(G′)

f ′(z)

for every x ∈ V (G). We now consider div(φ∗f ′)(x). We have

div(φ∗f ′)(x) = div(f ′ φ)(x) = deg(x)f ′(φ(x))−∑

e=xy∈E(G)

f ′(φ(y)).

By (2.2), we have

deg(x)f ′(φ(x)) = mφ(x) deg(φ(x))f ′(φ(x)) +∑

e=xy∈E(G), φ(y)=φ(x)

f ′(φ(y)).

Therefore

(4.8) div(φ∗f ′)(x) = mφ(x) deg(φ(x))f ′(φ(x))−∑

e=xy∈E(G), φ(y) 6=φ(x)

f ′(φ(y)).

Moreover, for every edge e′ = zφ(x) ∈ E(G′) we have∑e=xy, φ(e)=e′

f ′(φ(y)) = mφ(x)f ′(z),

and therefore ∑e=xy∈E(G), φ(y) 6=φ(x)

f ′(φ(y)) = mφ(x)∑

e′=zφ(x)∈E(G′)

f ′(z).

Thus (4.4) follows from (4.7) and (4.8).

In particular:

Corollary 4.9. If φ : G → G′ is a harmonic morphism, then φ∗(Prin(G)) ⊆ Prin(G′) andφ∗(Prin(G′)) ⊆ Prin(G).

As a consequence of Corollary 4.9, we see that φ induces group homomorphisms (which wecontinue to denote by φ∗, φ∗)

φ∗ : Jac(G) → Jac(G′), φ∗ : Jac(G′) → Jac(G).

It is straightforward to check that if ψ : G → G′ and φ : G′ → G′′ are harmonic morphisms andD ∈ Div(G), D′′ ∈ Div(G′′), then φψ : G→ G′′ is harmonic, and we have (φψ)∗(D) = φ∗(ψ∗(D))and (φ ψ)∗(D′′) = ψ∗(φ∗(D′′)). Therefore we obtain two different functors from the category ofgraphs (together with harmonic morphisms between them) to the category of abelian groups: acovariant “Albanese” functor (G 7→ Jac(G), φ 7→ φ∗) and a contravariant “Picard” functor (G 7→Jac(G), φ 7→ φ∗). (This terminology comes from the corresponding notions in algebraic geometry.)

Remark 4.10. A more conceptual definition of the Albanese functor φ∗ is as follows. Choose abase vertex x0 ∈ G, and let S = Sx0 : G → Jac(G) and S′ = Sφ(x0) : G′ → Jac(G′) denote thecorresponding Abel-Jacobi maps. Since S′ : G′ → Jac(G′) is a harmonic function, it follows fromProposition 2.8 that the pullback S′ φ is a harmonic map from G to Jac(G′). As S′ φ sends x0

to 0, it follows from Lemma 1.8 that there is a unique homomorphism ψ : Jac(G) → Jac(G′) suchthat S′ φ = ψ S. From the uniqueness of ψ, it follows easily that ψ = φ∗.


In particular, a harmonic morphism φ : G→ G′ gives rise to a commutative diagram

Gφ−−−−→ G′

S

y yS′

Jac(G)φ∗−−−−→ Jac(G′)

As an application of Corollary 4.9, we have the following result:

Corollary 4.11. If φ : G→ G′ is a non-constant harmonic morphism, then for every D ∈ Div(G)we have rG′(φ∗(D)) ≥ rG(D).

Proof. By Lemma 2.7, φ is surjective on vertices. Let D′ := φ∗(D), and let k be a nonnegativeinteger. For every effective divisor E′ ∈ Div(G′) of degree k, we can choose E ∈ Div(G) such thatφ∗(E) = E′. If rG(D) ≥ k, then by definition D − E = F + P with F effective and P principal.Applying φ∗ and using the fact that φ∗(P ) ∈ Prin(G′), we see that D′ − E′ is equivalent to theeffective divisor φ∗(F ), and therefore rG′(D′) ≥ k as well.

We now investigate some useful general properties of the induced maps φ∗ and φ∗ on Jacobians.In the classical algebraic geometry setting, φ∗ is always surjective but φ∗ is sometimes injective andsometimes not. More precisely, φ∗ : Jac(X ′) → Jac(X) is injective if and only if φ : X → X ′ hasa nontrivial unramified abelian subcover. The situation for graphs is simpler, since as we will nowshow, φ∗ is always surjective and φ∗ is always injective. The surjectivity of φ∗ is easy:

Lemma 4.12. Let φ : G→ G′ be a non-constant harmonic morphism. Then φ∗ : Jac(G) → Jac(G′)is surjective.

Proof. It follows from Lemma 2.7 and the linearity of φ∗ that φ∗ is a surjective map from Div(G)to Div(G′), which implies surjectivity on the level of Jacobians.

The injectivity of φ∗ is much more subtle (as one would expect, since the analogous statementfor Riemann surfaces is false):

Theorem 4.13. Let φ : G → G′ be a non-constant harmonic morphism. Then φ∗ : Jac(G′) →Jac(G) is injective.

Proof. We first set the following notation. For a function f : V (G) → Z, let max(f) = maxx∈V (G) f(x),let min(f) = minx∈V (G) f(x), and let s(f) = max(f) − min(f). Let M(f) = x ∈ V (G) |f(x) =max(f), and let m(f) = x ∈ V (G) |f(x) = min(f).

It suffices to show that D′ ∈ Prin(G′) for every D′ ∈ Div(G′) such that φ∗(D′) ∈ Prin(G).Suppose for the sake of contradiction that there exists a divisor D′ ∈ Div(G′) r Prin(G′) such thatφ∗(D′) = div(f) for some f : V (G) → Z. Choose such a D′ for which s(f) is minimized, and subjectto this condition such that |M(f)| is minimized. Let D := φ∗(D′) = div(f). Clearly s(f) 6= 0, asotherwise D = 0, and therefore D′ = 0 ∈ Prin(G′), a contradiction. Therefore there exists a vertexx0 ∈M(f) with a neighbor in V (G) rM(f). For every x ∈M(f), one has

D(x) = div(f)(x) ≥ |e ∈ E(G) | e = xy, y ∈ V (G) rM(f)|.It follows that D(x) ≥ 0 for every x ∈ M(f), and that D(x0) > 0. Similarly, for every x ∈ m(f)one has either D(x) < 0, or else D(x) = 0 and all the neighbors of x belong to m(f). LetX = φ−1(φ(x0))∩m(f). Since D(x0) > 0, we have D′(φ(x0)) > 0 as well, so by the definition of φ∗

it follows that D(x) > 0 for every x ∈ φ−1(φ(x0)) with mφ(x) > 0. Therefore X consists entirely


of vertices x ∈ φ−1(φ(x0)) with mφ(x) = 0 and D(x) = 0. But then all the neighbors of vertices inX belong to φ−1(φ(x0)), and thus by the above must belong to X. Since G is connected, it followsthat X is empty, i.e.,

(4.14) φ−1(φ(x0)) ∩m(f) = ∅.

Let χ : V (G′) → Z be the characteristic function of φ(x0), and let D′′ = D′ − div(χ). Weclaim that D′′ contradicts the choice of D′. Clearly, D′′ ∈ Div(G′) r Prin(G′). By Proposition 4.2,we have

φ∗(D′′) = φ∗(D′)− φ∗(div(χ)) = div(f)− div(φ∗χ) = div(f − χ φ).

Let D? = φ∗(D′′) and let f? = f − χ φ. We have f?(x) = f(x)− 1 for every x ∈ φ−1(φ(x0)), andf?(x) = f(x) otherwise. By (4.14), we have min(f) = min(f?), and clearly max(f) ≥ max(f?).Therefore s(f) ≥ s(f?). Moreover, either s(f) > s(f?) or max(f) = max(f?). In the second case,we have M(f?) ⊆ M(f) r x0, and thus |M(f)| > |M(f?)|. It follows that D′′ contradicts thechoice of D′, as claimed.

4.2. Eulerian cuts and harmonic morphisms. Let κG = | Jac(G)| denote the number of span-ning trees in a graph G. From either Lemma 4.12 or Theorem 4.13, we immediately deduce thefollowing corollary:

Corollary 4.15. If there exists a non-constant harmonic morphism from G to G′, then κG′ dividesκG.

Define an Eulerian cut in a graph G to be a non-empty cut which is also an even subgraph of G;equivalently, an Eulerian cut is a partition of V (G) into non-empty disjoint subsets X and X ′ insuch a way that there are an even number of edges connecting each vertex inX (resp. X ′) to verticesin X ′ (resp. X). According to a theorem of Chen [Che71] (see also [Big97, Proposition 35.2]), Ghas an Eulerian cut if and only if κG is even. From Corollary 4.15, it therefore follows that if G′

has an Eulerian cut and there exists a non-constant harmonic morphism from G to G′, then Ghas an Eulerian cut as well. We can strengthen this observation using the following result, whichcharacterizes the existence of Eulerian cuts in terms of non-constant harmonic maps from G to thegraph B2 consisting of 2 vertices connected by 2 edges:

Theorem 4.16. Let G be a graph. Then the following are equivalent:(1) G has an Eulerian cut.(2) There is a non-constant harmonic morphism from G to B2.(3) κG is even.

Proof. Although the equivalence (1) ⇔ (3) is just Chen’s theorem, for the reader’s conveniencewe will provide a self-contained proof of this result. Our proof of (3) ⇒ (1) is borrowed from theunpublished manuscript [Epp96].

(1) ⇒ (2) : Suppose that G admits an Eulerian cut S. We claim that there exists a partition ofS into non-empty disjoint subsets S1, S2 ⊆ E(G) such that

|e ∈ S1 | x ∈ e| = |e ∈ S2 | x ∈ e|

for every x ∈ V (G). Indeed, it is well-known (see [Bol98, §I.1,Theorem 1]) that the edge set of anyEulerian graph can be decomposed into edge-disjoint cycles. Since the graph G[S] with vertex setV (G) and edge set S is Eulerian and bipartite, it follows that G[S] decomposes into edge-disjointeven cycles. It is trivial to obtain the required partition for an even cycle. By composing theresulting partitions of even cycles, one then obtains the required partition (S1, S2) of S.


We now construct a non-constant harmonic morphism φ : G → B2 as follows. Let the verticesof B2 be labeled x and y, and let the edges of B2 be labeled e1 and e2. Let X ⊆ V (G) be one ofthe sides of S. For z ∈ V (G), let φ(z) = x if x ∈ X, and let φ(z) = y otherwise. For i ∈ 1, 2 ande ∈ Si, let φ(e) = ei. Finally, if e = z1z2 ∈ E(G) r S, we set φ(e) = φ(z1) = φ(z2). It follows fromthe definition of S1 and S2 that φ is a non-constant harmonic morphism.

(2) ⇒ (3) : We have κB2 = 2. Therefore, if G admits a non-constant harmonic morphism to B2,then κG is even by Corollary 4.15.

(3) ⇒ (1) : (c.f. [Epp96, Proof of Theorem 6]) Let Λ(G) = H1(G,Z) ⊂ H1(G,R) denote thelattice of integral flows on G, and let Λ#(G) be the lattice dual to Λ(G) under the standardEuclidean inner product 〈 , 〉 on C1(G,R) ⊇ H1(G,R). Explicitly, we have

Λ# = ω ∈ H1(G,R) | 〈ω, ω′〉 ∈ Z for all ω′ ∈ H1(G,Z).

By [BdlHN97] (see also [Big97, §29]), there is a canonical isomorphism Jac(G) ∼= Λ#(G)/Λ(G).Suppose that κG is even. Then Jac(G) has an element of order 2, so there is a flow ω ∈ Λ#(G)

such that ω 6∈ Λ(G) but 2ω ∈ Λ(G). Thus the value of ω along each edge of G is a half-integer,and the set S of edges along which ω is non-integral is non-empty. Since δ(ω) = 0, it follows thatevery vertex in S has even degree. So it suffices to prove that S is a cut. To see this, choose avertex x ∈ V (G), and partition V (G) into disjoint subsets A and B as follows: a vertex y ∈ V (G)belongs to A (resp. B) iff it can be connected to x by a path containing an odd (resp. even) numberof edges in S. Since G is connected, A ∪ B = V (G). Furthermore, we have A ∩ B = ∅, becauseotherwise there would be a cycle C in G containing an odd number of edges of S, and therefore〈ω, χC〉 6∈ Z, contradicting the fact that ω ∈ Λ#(G). Finally, to see that S is indeed a cut, notethat each edge e ∈ S connects a vertex in A to a vertex in B (since e itself is a path with one edgein S), and an edge e′ 6∈ S cannot connect a vertex in A to a vertex in B (since e′ is a path with noedges in S). Thus S is precisely the cut consisting of all edges connecting A to B.

Remark 4.17. Here is a more direct argument for proving (1) ⇒ (3) which makes use of Theorem 1.5.Let S be an Eulerian cut in G separating the subsets X,Y ⊂ V (G). It is easy to see that thereexists an ordering x1, . . . , xk of X such that for every i ∈ 1, . . . , k either outdegX(xi) > 0 orxjxi ∈ E(G) for some j < i. Similarly, there exists an ordering y1, . . . , y` of Y such that forevery i ∈ 1, . . . , l either outdegY (yi) > 0 or yjyi ∈ E(G) for some j < i. Define a divisorD ∈ Div0(G) by setting D(x) := 1

2 outdegX(x) for x ∈ X, and D(y) := −12 outdegY (y) for y ∈ Y .

Then 2D = div(χX) ∼ 0. However, using Theorem 1.5, we see that D itself is not equivalent to 0,since D ≤ ν, where ν is the divisor associated to the linear order y1 < · · · < y` < x1 < · · · < xk onV (G). Thus D corresponds to an element of order 2 in Jac(G), and in particular κG = | Jac(G)| iseven.

4.3. Induced maps on harmonic 1-forms. We now turn to a discussion of harmonic 1-formsand the maps induced on them by a harmonic morphism.

We begin with some notation and terminology. Let ~E(G) denote the set of directed edges of G.For e ∈ ~E(G), we let o(e), t(e) denote the origin and terminus of e, respectively. We denote by ethe directed edge representing the same undirected edge as e, but with the opposite orientation.From the definition of a morphism of graphs, it follows easily that a morphism φ : G→ G′ inducesa natural map from ~E(G) to ~E(G′) ∪ V (G′).

Let A be an abelian group, and let C1(G,A) denote the space of 1-cochains on G with valuesin A, i.e., functions ω : ~E(G) → A with the property that ω(e) = −ω(e) for all e ∈ ~E(G). As


usual, we also let C0(G,A) denote the space of all functions f : V (G) → A. We define an operatorδ : C1(G,A) → C0(G,A) by the formula

(4.18) δ(ω)(x) :=∑

e∈ ~E(G)t(e)=x

ω(e).

An A-flow (or simply a flow if A = R) on G is a 1-cochain ω ∈ C1(G,A) such that δ(ω) = 0.We denote by H1(G,A) the space of A-flows on G. When A = R, we will also refer to H1(G) :=H1(G,R) as the space of harmonic 1-forms on G; it is analogous to the space Ω1(X) of holomorphic1-forms on a Riemann surface X. For example, it is well-known that dimR H1(G) = g (just asdimC Ω1(X) = g in the Riemann surface case).

Suppose φ : G → G′ is a harmonic morphism and that ω ∈ C1(G,A), ω′ ∈ C1(G′, A) are1-cochains. We define the pullback φ∗ω′ ∈ C1(G,A) by

(φ∗ω′)(e) :=ω′(φ(e)) if φ(e) ∈ ~E(G′)0 otherwise

and the push-forward (or trace) φ∗ω ∈ C1(G′, A) by

φ∗ω(e′) :=∑

e∈ ~E(G)φ(e)=e′

ω(e).

Proposition 4.19. Let φ : G → G′ be a harmonic morphism and let ω ∈ H1(G), ω′ ∈ H1(G′) beharmonic 1-forms. Then:

(1) φ∗ω′ ∈ H1(G).(2) φ∗ω ∈ H1(G′).

Proof. To establish (1), we follow [Ura00, Proof of Theorem 2.13]. For every x ∈ V (G), we have∑e∈ ~E(G),t(e)=x

φ(e)∈ ~E(G′)

ω′(φ(e)) =∑

e′∈ ~E(G′)t(e′)=φ(x)

∑e∈ ~E(G),x∈e

φ(e)=e′

ω′(φ(e)) = mφ(x)∑

e′∈ ~E(G′)t(e′)=φ(x)

ω′(e′).

Since δ(ω′) = 0 and (φ∗ω′)(e) = 0 for all vertical edges e ∈ ~E(G), for all x ∈ V (G) we have

δ(φ∗ω′)(x) =∑

e∈ ~E(G)t(e)=x

(φ∗ω′)(e) =∑

e∈ ~E(G),t(e)=x

φ(e)∈ ~E(G′)

ω′(φ(e))

= mφ(x)∑

e′∈ ~E(G′)t(e′)=φ(x)

ω′(e′) = mφ(x)δ(ω′)(φ(x))

= 0,

which proves (1).For (2), note that for every y ∈ V (G′), we have

(4.20)∑

e′∈ ~E(G′)t(e′)=y

∑e∈ ~E(G)φ(e)=e′

ω(e) =∑

x∈V (G)φ(x)=y

∑e∈ ~E(G)t(e)=x

ω(e),


since each vertical edge in E(G) gets counted twice in the sum on the right-hand side of (4.20),once with each orientation, and therefore the net contribution to the sum from such an edge is zero.Therefore

δ(φ∗ω)(y) =∑

e′∈ ~E(G′)t(e)=y

(φ∗ω)(e′) =∑

e′∈ ~E(G′)t(e′)=y

∑e∈ ~E(G)φ(e)=e′

ω(e)

=∑

x∈V (G)φ(x)=y

∑e∈ ~E(G)t(e)=x

ω(e) by (4.20)

=∑

x∈V (G)φ(x)=y

δ(ω)(x) = 0,

proving (2).

As a consequence of Proposition 4.19, we see that φ induces linear transformations (which wecontinue to denote by φ∗, φ∗)

φ∗ : H1(G′) → H1(G), φ∗ : H1(G) → H1(G′).

It is straightforward to check that the association (G′, φ) 7→ (H1(G′), φ∗) (resp. (G,φ) 7→ (H1(G), φ∗))is a contravariant (resp. covariant) functor from the category of graphs (together with harmonicmorphisms between them) to the category of vector spaces.

It follows easily from the definitions that

(4.21) φ∗φ∗(ω′) = deg(φ)ω′

for all ω′ ∈ H1(G′) (compare with Lemma 4.1). As a consequence, we obtain the following result,which provides another way to see that if φ is a non-constant harmonic morphism from a graph ofgenus g to a graph of genus g′, then g′ ≤ g (c.f. Theorem 2.14):

Corollary 4.22. If φ : G→ G′ is a non-constant harmonic morphism, then φ∗ : H1(G′) → H1(G)is injective and φ∗ : H1(G) → H1(G′) is surjective.

Proof. Both the injectivity of φ∗ and the surjectivity of φ∗ follow easily from (4.21). However, onecan also prove the injectivity of φ∗ directly (c.f. [Ura00, Proof of Theorem 2.13]): if φ∗(ω′) = 0,then ω′(φ(e)) = 0 for all horizontal edges e ∈ E(G), and since φ maps the set of horizontal edgesof G surjectively onto E(G′), it follows that ω′ = 0.

By functoriality, an automorphism α of a graph G induces an automorphism α∗ of the vectorspace H1(G). For later use, we note the following property of the corresponding map Aut(G) →Aut(H1(G)):

Proposition 4.23. If G is a 2-edge-connected graph of genus at least 2, then the natural map fromAut(G) to Aut(H1(G)) is injective.

Proof. Let β, β′ ∈ Aut(G). By considering the automorphism α := β′β−1, it suffices to prove thatif α∗ is the identity map on H1(G), then α is the identity map on G. So suppose α∗ = Id. Thenevery directed cycle in G is mapped onto itself. Let C be an (undirected) simple cycle in G (i.e., acycle with no repeated vertices), let x ∈ C be a vertex of degree at least 3, and let x′ = α(x). Lete ∈ C be a directed edge with o(e) = x, let e′ = α(e) ∈ C, and let e′′ ∈ ~E(G) be a directed edgewith o(e′′) = x and e′′ 6∈ C. Since G is 2-edge-connected, e′′ belongs to a simple cycle C ′′, and we


can choose C ′′ so that either V (C) ∩ V (C ′′) = x, or else so that E(C) ∩ E(C ′′) is a path in Ccontaining e.

Case I: V (C) ∩ V (C ′′) = x.In this case, x′ ∈ V (C) ∩ V (C ′′) = x so α(x) = x. But then α(e) = e, since α∗ preserves

directed cycles of G. From this it follows easily that α is the identity map on C.Case II: E(C) ∩ E(C ′′) is a path in C containing e.In this case, we must also have e′ ∈ C ′′. Suppose e′ 6= e. Then as α(e′′) 6∈ C, the cycle C ′′ can

be directed so that it consists of the unique path in C from x to x′ followed by the unique pathin C ′′\C from x′ to x. But then α restricted to C ′′ is orientation-reversing, a contradiction. Weconclude that α(e) = e, and hence α is the identity map on C in this case as well.

It follows that the restriction of α to every simple cycle C of G is the identity map. Since G is2-edge-connected, this implies that α is the identity map on all of G.

Remark 4.24. Proposition 4.23 is the analogue of the fact from algebraic geometry that if X is aRiemann surface of genus at least 2, then the natural map from Aut(X) to Aut(Ω1(X)) is injective.

As a consequence of Proposition 4.23, we obtain the following non-trivial restriction on theautomorphism group of a 2-edge-connected graph of genus at least 2:

Corollary 4.25. If G is a 2-edge-connected graph of genus g ≥ 2, then the group Aut(G) isisomorphic to a subgroup of the group GL(g,Z) of invertible g × g matrices with coefficients in Z.

Proof. Since Aut(G) acts faithfully on the g-dimensional vector space H1(G,R) and preserves thelattice H1(G,Z), the result follows.

Remark 4.26. By a theorem of Minkowski,if H is a finite subgroup of GL(n,Z), then every primedivisor p of |H| satisfies p ≤ n+ 1. In particular, Corollary 4.25 implies that if a 2-edge-connectedgraph G of genus g ≥ 2 has an automorphism of prime order p then p ≤ g + 1. This boundis sharp, since the graph Bn+1 consisting of 2 vertices joined by n + 1 edges has genus n and|Aut(Bn+1)| = 2(n+ 1)!.

5. Hyperelliptic graphs

5.1. Definition and basic properties. We say that a graph G is hyperelliptic if there exists adivisor D ∈ Div(G) such that deg(D) = 2 and r(D) = 1. By Riemann-Roch for graphs, if G ishyperelliptic then g(G) ≥ 2, and by Clifford’s theorem for graphs, if g(G) ≥ 2 and deg(D) = 2,then r(D) = 1 if and only if r(D) ≥ 1.

Example 5.1. Every graph of genus 2 is hyperelliptic. Indeed, if g(G) = 2, then the canonicaldivisor KG has deg(KG) = 2, and r(KG) = 1 by Riemann-Roch for graphs.

Example 5.2. Let the graph G = B(l1, l2, . . . , ln) consist of two vertices x and y and n ≥ 3internally disjoint paths joining x to y with lengths l1, l2, . . . , ln. Then G is hyperelliptic. Morespecifically, we claim that r((x)+(y)) = 1. To prove this, it suffices to show that |(x)+(y)−(z)| 6= ∅for every z ∈ V (G). Consider one of the paths joining x and y, and let x, z1, z2, . . . , zl−1, y be thevertices of this path in order. Then (x) + (y) ∼ (zi) + (zl−i), and therefore |(x) + (y)− (zi)| 6= ∅ forevery 1 ≤ i ≤ l − 1. Thus r((x) + (y)) = 1, and our claim follows.

Although the graph G = B(1, 1, . . . , 1) has edge connectivity equal to |E(G)|, which can be arbi-trarily large, the following result shows that every other hyperelliptic graph has edge connectivityat most 2:


Lemma 5.3. If G is a hyperelliptic graph, then either |V (G)| = 2 (so that G is isomorphic to agraph of the form B(1, 1, . . . , 1)) or G has edge connectivity at most 2.

Proof. Let D = (x) + (x′) be an effective divisor of degree 2 on G with r(D) = 1. If |V (G)| > 2,choose a vertex y ∈ V (G) with y 6∈ x, x′. Since r(D) = 1, there exists y′ ∈ V (G) such that(x) + (x′) ∼ (y) + (y′), and therefore the map S(2) : Div2

+(G) → Jac(G) is not injective. ByTheorem 1.9, it follows that G is not 3-edge-connected.

A classical result from algebraic geometry asserts that if X is a hyperelliptic Riemann surfaceand φ : X → X ′ is a non-constant holomorphic map with g(X ′) ≥ 2, then X ′ is also hyperelliptic.Using Corollary 4.11, we obtain the following analogous result for graphs:

Corollary 5.4. If G is hyperelliptic and φ : G→ G′ is a non-constant harmonic morphism onto agraph G′ with g(G′) ≥ 2, then G′ is hyperelliptic as well.

As in classical algebraic geometry, we can also show in the graph-theoretic setting that there isat most one complete linear system |D| of degree 2 on a graph G for which r(D) = 1:

Proposition 5.5. If D,D′ are degree 2 divisors on G with r(D) = r(D′) = 1, then D ∼ D′.

Proof. Let g := g(G). Consider the divisor E := D + (g − 2)D′ of degree 2g − 2 on G. ByLemma 1.2, we have r(E) ≥ g − 1. By Riemann-Roch for graphs, we have r(KG − E) ≥ 0; sincedeg(KG − E) = 0, it follows that KG ∼ E. Applying the same reasoning to E′ := (g − 1)D′, wesee that KG ∼ E′, and therefore D ∼ D′ as desired.

5.2. Hyperelliptic graphs, involutions, and harmonic morphisms. Our next goal is to ob-tain a graph-theoretic analogue of the well-known result from algebraic geometry that the followingare equivalent for a Riemann surface X of genus at least 2: (i) X is hyperelliptic; (ii) X admits anon-constant holomorphic map of degree 2 onto the Riemann sphere; and (iii) there is an involutionι : X → X whose quotient is isomorphic to the Riemann sphere. We begin by discussing quotientsin the category of graphs (together with morphisms between them).

Let H be a finite group acting on a graph G, i.e., suppose we are given a homomorphismH → Aut(G). We write h · x for the action of an element h ∈ H on an element x of V (G) ∪E(G).We define the quotient graph G/H, together with a canonical morphism πH : G→ G/H, as follows.

For x, y ∈ V (G) ∪ E(G), let x ∼H y if there exists an element h ∈ H such that h · x = y. Then∼H is an equivalence relation on V (G)∪E(G). The quotient graph G/H is constructed as follows.The vertices of G/H are the equivalence classes of V (G) with respect to ∼H . The edges of G/Hcorrespond to those equivalence classes of E(G) with respect to ∼H which consist of edges whoseends are inequivalent. It is readily verified that G/H is a graph in our sense of the word (i.e., aconnected multigraph with no loop edges). The quotient morphism πH : G → G/H maps everyvertex of G to its equivalence class, every edge of G whose ends are inequivalent to the edge of G/Hcorresponding to its equivalence class, and every edge of G with equivalent ends to the equivalenceclass of its ends. It is straightforward to check that πH is a surjective morphism of graphs (thoughnot necessarily a harmonic morphism), and by construction we have πH(h · x) = πH(x) for allh ∈ H and all x ∈ V (G)∪E(G). In fact, the morphism πH : G→ G/H has the following universalproperty: if π′ : G → G′ is any morphism of graphs for which π′(h · x) = π′(x) for all h ∈ H andall x ∈ V (G) ∪ E(G), then there exists a unique morphism ψ : G/H → G′ such that π′ = ψ πH .This universal property uniquely characterizes G/H up to isomorphism.


If H = 〈φ〉 is a cyclic subgroup of Aut(G), we will often write G/φ instead of G/H and φ∼

instead of πφ.

An automorphism ι of a graph G is called an involution if ι ι is the identity automorphism.We say that an involution ι is mixing if for every edge e = xy ∈ E(G) such that ι(e) = e wehave ι(x) = y. Equivalently, ι is mixing if and only if it does not fix any directed edge of G. Thefollowing lemma shows that if |V (G)| > 2, there is a one-to-one correspondence between mixinginvolutions of G and non-degenerate harmonic morphisms of degree two from G to a graph G′.

Lemma 5.6. Let G,G′ be graphs, and let φ : G→ G′ be a non-degenerate harmonic morphism ofdegree 2. Then there is a mixing involution ι of G for which φ = ι∼. Conversely, let |V (G)| > 2and let ι : G → G be a mixing involution. Then ι∼ is a non-degenerate harmonic morphism ofdegree two.

Proof. Let φ : G→ G′ be a non-degenerate harmonic morphism of degree 2. For x ∈ V (G), if thereexists y 6= x such that φ(y) = φ(x) then we define ι(x) = y. Otherwise, we define ι(x) = x. Forevery e ∈ E(G) such that φ(e) ∈ E(G′), there is a unique edge e′ ∈ E(G) such that e′ 6= e andφ(e′) = φ(e), and we define ι(e) = e′. Define ι(e) = e for every e ∈ E(G) such that φ(e) ∈ V (G′).

If x ∈ V (G), e ∈ E(G), x ∈ e and φ(e) ∈ E(G′) then either ι(x) ∈ ι(e), or x ∈ ι(e). In the secondcase, mφ(x) = 2 and therefore by non-degeneracy of φ we have x = ι(x). It follows easily from thisthat ι is a morphism. Clearly ι ι is the identity map. In particular, ι is a bijection. Therefore ι isan involution, and ι is mixing by definition. Finally, it is easy to see that φ = ι∼.

Now suppose |V (G)| > 2, and let ι : G → G be a mixing involution of G. Denote G/ι by G′.Note that |V (G′)| ≥ |V (G)|/2 > 1. Consider a vertex x ∈ V (G), let y = ι∼(x), and consideran edge e′ = yy′ ∈ E(G′). Then there exists an edge e = xx′ in G such that ι∼(e) = e′, and(ι∼)−1(e′) = e, ι(e). Therefore |d ∈ E(G)|x ∈ d, ι∼(d) = e′| = 1 if x 6= ι(x) and |d ∈E(G)|x ∈ d, ι∼(d) = e′| = 2 otherwise. It follows that mι∼(x) is well defined and positive,and that

∑ι∼(y)=z mι∼(y) = 2 for every z ∈ V (G′). Therefore, ι∼ is a non-degenerate harmonic

morphism of degree two, as claimed.

The following result will be used to reduce the study of general hyperelliptic graphs to the specialcase of graphs which are 2-edge-connected.

Lemma 5.7. Let G be a graph, let G be the graph obtained by contracting every bridge of G, andlet ρ : G → G be the natural surjective morphism. Then for every divisor D ∈ Div(G), we haveD ∼G 0 if and only if ρ∗(D) ∼G 0, where ρ∗(D) is defined as in (2.11).

Remark 5.8. Note that the morphism ρ : G→ G is not necessarily harmonic, c.f. Example 3.6.

Proof of Lemma 5.7. It suffices by induction to prove the result with G replaced by the graphobtained by contracting a single bridge e. We begin with some notation. Let x1, x2 be the endpointsof e, and let x = ρ(x1) = ρ(x2). Let G1, G2 be the connected components of G − e containing x1

and x2, respectively, and for i = 1, 2, let Gi = ρ(Gi), so that G = G1 ∪ G2 and G1 ∩ G2 = x.Note that (x1) ∼ (x2) on G; this follows from the observation that (x1)− (x2) = div(χG1).

Let D ∈ Div(G). Suppose first that D is a principal divisor on G; we want to show that ρ∗(D)is a principal divisor on G. It suffices by linearity to consider the case where D = div(χy) for somey ∈ V (G). If y 6∈ x1, x2, then ρ∗(D) = div(χρ(y)). Otherwise, we have ρ∗(div(χx1)) = div(χV (G2))and ρ∗(div(χx2)) = div(χV (G1)). This proves that ρ∗(D) is principal.

In the other direction, suppose that ρ∗(D) is principal; we want to show that D itself is principal.By linearity, it suffices to consider the case where ρ∗(D) = div(χz) for some z ∈ V (G). If z 6= x,


then ρ−1(z) consists of a single element, and D = div(χρ−1(z)). If z = x, then ρ−1(z) = x1, x2and using the fact that (x1) ∼ (x2) it is easy to see that D ∼ div(χx1,x2). This proves that D isprincipal.

Remark 5.9. As alluded to in [BN07, Remark 4.8], one can use Lemma 5.7 to obtain an alternateproof of Corollary 4.7 from [BN07] which does not make use of circuit theory.

Corollary 5.10. Let G be a graph, let G be the graph obtained by contracting every bridge of G,and let ρ : G→ G be the natural surjective morphism. Then for every divisor D ∈ Div(G), we haverG(D) = rG(ρ∗(D)).

Proof. Let k ≥ 0 be an integer, and let D ∈ Div(G). Suppose r(D) ≥ k, and let D = ρ∗(D). Thenfor every effective divisor E ∈ Div(G) of degree k, there exists an effective divisor E′ ∈ Div(G)such that D − E ∼ E′, and thus D − ρ∗(E) ∼ ρ∗(E′) by Lemma 5.7. Since ρ∗ : Div(G) → Div(G)is surjective and preserves degrees and effectivity, it follows that r(D) ≥ k.

Conversely, suppose r(ρ∗(D)) ≥ k. Then for every effective divisor E ∈ Div(G) of degree k,there exists an effective divisor E′ ∈ Div(G) such that ρ∗(D)− ρ∗(E) ∼ ρ∗(E′). By Lemma 5.7, itfollows that D − E ∼ E′, and thus r(D) ≥ k as desired.

Corollary 5.11. Let G be a graph, and let G be the graph obtained by contracting every bridge ofG. Then G is hyperelliptic if and only if G is hyperelliptic.

Proof. This follows immediately from Corollary 5.10 and the surjectivity of ρ∗ : Div(G) → Div(G).

Because of Corollary 5.11, when studying hyperelliptic graphs there is no loss of generality ifwe restrict our attention to graphs which are 2-edge-connected. And it turns out that for 2-edge-connected graphs, there are several equivalent characterizations of what it means to be hyperelliptic:

Theorem 5.12. For a 2-edge-connected graph G of genus g ≥ 2, the following conditions areequivalent:

(1) G is hyperelliptic.(2) There exists an involution ι : G→ G such that G/ι is a tree.(3) There exists a non-degenerate degree two harmonic morphism φ from G to a tree, or

|V (G)| = 2.

Proof. If |V (G)| = 2 then it is easily verified that conditions (1), (2) and (3) all hold. Therefore inwhat follows we assume |V (G)| > 2.

(1) ⇒ (2). Let D be a divisor of degree 2 on G with r(D) = 1. For every x ∈ V (G), we have|D − (x)| 6= ∅ and deg(D − (x)) = 1. Since G is 2-edge-connected, by Theorem 1.9 there exists aunique y ∈ V (G) such that D − (x) ∼ (y). Define ι(x) = y.

Our next goal is to define ι on E(G). Consider an edge e = xy ∈ E(G). If ι(x) = y, we defineι(e) = e. If ι(x) 6= y, then let D1 = (x) + (ι(x)) and let D2 = (y) + (ι(y)). By the definition of ι,we have D1 ∼ D ∼ D2. Therefore, there exists a non-constant function f : V (G) → Z such thatD1 − D2 = div(f). Let M(f) be the set of all the vertices z ∈ V (G) for which f(z) is maximal.For every vertex z ∈M(f), we have

D1(z) ≥ (div(f))(z) =∑

e′=zz′∈E(G)

(f(z)− f(z′)) ≥ |e′ = zz′ ∈ E(G) | z′ ∈ V (G) rM(f)|.

Therefore deg(D1) ≥ |δ(M(f))|, where for X ⊆ V (G) we denote by δ(X) the set of all edges ofG having exactly one end in X. On the other hand, |δ(M(f))| ≥ 2 by the 2-edge connectivity of


G. It follows that |δ(M(f))| = 2, and that x, ι(x) ∈M(f). Analogously, we can conclude that f isminimized on y and ι(y), and therefore that y, ι(y) ∈ V (G) rM(f). It follows that e ∈ δ(M(f)).Define ι(e) to be the unique edge e∗ such that δ(M(f)) = e, e∗. Let x′ be the end of e∗ in M(f).By the argument above we have D1 = (x′) + (x). Therefore x′ = ι(x). By the symmetry betweenx and y, we conclude that e∗ joins ι(x) and ι(y). Therefore ι is an automorphism, and clearly ι ιis the identity.

By Lemma 5.6, we know that φ = ι∼ is a harmonic morphism. For every u,w ∈ V (G/ι) we have

φ∗((u)) = (u) + (ι(u)) ∼ D ∼ (w) + (ι(w)) = φ∗((w)).

Therefore, by Theorem 4.13, we have (u) ∼ (w) for all u,w ∈ V (G/ι). It follows from Lemma 1.1that G/ι is a tree, as desired.

(2) ⇔ (3). Consider an involution ι satisfying (2). For every edge e = xy ∈ E(G) such thatx 6= ι(y), the set of edges e, ι(e) is the preimage of an edge of G/ι, and therefore forms a cut inG. It follows that e 6= ι(e), and therefore ι is mixing. The equivalence of (2) and (3) now followsfrom Lemma 5.6.

(3) ⇒ (1). Let φ : G → T be a non-degenerate harmonic morphism of degree two, where T isa tree. Let y0 ∈ V (T ) be chosen arbitrarily and let D := φ∗((y0)). Then D is an effective divisorof degree 2 on G. We claim that r(D) = 1. Clearly, r(D) ≤ 1. Therefore, it suffices to show that|D − (x)| 6= ∅ for every x ∈ V (G). Note that (y) ∼ (y′) for every pair of vertices y, y′ ∈ V (T ).Therefore (φ(x)) ∼ (y0), and by Proposition 4.2 we have D ∼ φ∗((φ(x))) ≥ mφ(x)(x). By since φis non-degenerate, we have mφ(x) > 0, and therefore φ∗((φ(x))) = (x) + (x′) for some x′ ∈ V (G),which implies that |D − (x)| 6= ∅ as desired.

Remark 5.13. One can use Theorem 5.12 to give an alternate proof of Lemma 5.3 which does notmake use of Theorem 1.9. Indeed, if G is 2-edge-connected and |V (G)| > 2, then by Theorem 5.12there is a non-degenerate harmonic morphism φ of degree 2 from G to a tree T with |E(T )| > 0. Ife′ ∈ E(T ) and e, ι(e) are the distinct edges of G mapping to e′ under φ, then it is easy to see thatG− e, ι(e) is disconnected. Thus G is not 3-edge-connected.

It is worth stating explicitly the following fact which was established during the course of ourproof of Theorem 5.12:

Corollary 5.14. If G is a 2-edge-connected hyperelliptic graph, then for any involution ι for whichG/ι is a tree, we have (x)+(ι(x)) ∼ (y)+(ι(y)) for all x, y ∈ V (G). In particular, r((x)+(ι(x))) = 1for all x ∈ V (G).

From Corollary 5.14 and Proposition 5.5, we obtain the following graph-theoretic result whosestatement does not involve harmonic morphisms at all:

Corollary 5.15. If G is a 2-edge-connected graph of genus at least 2, then there is at most oneinvolution ι of G whose quotient is a tree.

Proof. By Corollary 5.14, if ι is such an involution then r((x)+(ι(x))) = 1 for all x ∈ V (G). So if ιand ι′ are two such involutions, then (x)+(ι(x)) ∼ (x)+(ι′(x)) for all x ∈ V (G) by Proposition 5.5.Thus (ι(x)) ∼ (ι′(x)) for all x ∈ V (G). Since G is 2-edge-connected, it follows from Theorem 1.9that ι(x) = ι′(x) for all x ∈ V (G), i.e., ι = ι′.

If G is a 2-edge-connected hyperelliptic graph, we call the unique involution ι whose quotient isa tree the hyperelliptic involution on G.

Remark 5.16. Corollary 5.15 is the graph-theoretic analogue of the fact that the hyperellipticinvolution on a hyperelliptic Riemann surface is unique. We will give another proof of Corollary 5.15in Remark 5.20 below.


Remark 5.17. It follows from the proofs of Theorem 5.12 and Corollary 5.15 that if G is a 2-edge-connected hyperelliptic graph and r((x) + (y)) = 1 for some x, y ∈ V (G), then y = ι(x).

As a consequence of the uniqueness of the hyperelliptic involution, we obtain the followingcorollary:

Corollary 5.18. If G is a 2-edge-connected hyperelliptic graph with hyperelliptic involution ι, thenι belongs to the center of the group Aut(G).

Proof. Let τ ∈ Aut(G), and consider the automorphism ι′ := τ−1ιτ . It is easy to check that ι′ isan involution, and that τ induces an isomorphism from G/ι′ to G/ι, so that G/ι′ is a tree. ByCorollary 5.15, we have ι′ = ι, and therefore ι and τ commute, as desired.

5.3. Equivalent characterizations of the hyperelliptic involution. For a Riemann surfaceX of genus at least 2 and ι : X → X an automorphism, the following are equivalent: (i) X ishyperelliptic with hyperelliptic involution ι; (ii) ι∗ : Jac(X) → Jac(X) is multiplication by −1; (iii)ι∗ : Jac(X) → Jac(X) is multiplication by −1; (iv) ι∗ : Ω1(X) → Ω1(X) is multiplication by −1;and (v) ι∗ : Ω1(X) → Ω1(X) is multiplication by −1. We now show that a similar characterizationholds for 2-edge-connected graphs with genus at least 2.

Theorem 5.19. Let G be a 2-edge-connected graph of genus g ≥ 2, and let ι ∈ Aut(G). Then thefollowing are equivalent:

(1) G is hyperelliptic with hyperelliptic involution ι.(2) ι∗ : Jac(G) → Jac(G) is multiplication by −1.(3) ι∗ : Jac(G) → Jac(G) is multiplication by −1.(4) ι∗ : H1(G) → H1(G) is multiplication by −1.(5) ι∗ : H1(G) → H1(G) is multiplication by −1.

Proof. Since ι is a harmonic morphism of degree 1 from G to itself, ι∗ ι∗ is the identity map onboth Jac(G) and H1(G). It follows easily that (2) ⇔ (3) and (4) ⇔ (5). So it suffices to prove that(1) ⇔ (2) and (1) ⇔ (5).

(1) ⇒ (2). If G is hyperelliptic with hyperelliptic involution ι, then by Corollary 5.14, for everyx, y ∈ V (G), we have (x) + (ι(x)) ∼ (y) + (ι(y)). Thus (x)− (y) ∼ (ι(y))− (ι(x)) = ι∗((y)− (x)).Since the group Div0(G) is generated by divisors of the form (x)− (y), it follows that ι∗ ≡ −1 onJac(G).

(2) ⇒ (1). If ι∗ ≡ −1 on Jac(G), then (x)+(ι(x)) ∼ (y)+(ι(y)) for all x, y ∈ V (G). In particular,for any x ∈ V (G), we have r((x) + (ι(x))) = 1. Thus G is hyperelliptic. By Remark 5.17, ι is thehyperelliptic involution on G.

(1) ⇒ (5). Suppose G is hyperelliptic with hyperelliptic involution ι, and let π : G → T be thecorresponding quotient map from G to a tree T . If |V (G)| = 2, it is clear that (5) holds, so we mayassume that |V (T )| > 1. Let e′ ∈ ~E(T ) be a directed edge of T . Since π is a harmonic morphism ofdegree 2, there are two distinct directed edges e, ι(e) of G mapping onto e′. Let ω ∈ H1(G). SinceT is a tree, we have H1(T ) = 0, and therefore (π∗ω)(e′) = 0. On the other hand, by definition wehave

(π∗ω)(e′) = ω(e) + ω(ι(e)) = ω(e) + (ι∗ω)(e).

It follows that (ι∗ω)(e) = −ω(e) for all e ∈ ~E(G) such that π(e) ∈ ~E(T ). But for e ∈ ~E(G) withπ(e) ∈ V (T ), we have ι(e) = e, and thus (ι∗ω)(e) = −ω(e) for such edges as well. It follows that(ι∗ω)(e) = −ω(e) for all e ∈ ~E(G), as desired.


(5) ⇒ (1). Suppose ι∗ ≡ −1 on H1(G). Then (ι2)∗ is the identity map on H1(G), so ι isan involution by Proposition 4.23. If ι(e) = e for some directed edge e, then letting ω be thecharacteristic function of any simple cycle containing e, we have

ω(e) = ω(ι(e)) = (ι∗ω)(e) = −ω(e),

so that ω(e) = 0, a contradiction. Therefore ι is mixing. If |V (G)| = 2, it is easy to verify directlythat (1) holds. So we may assume without loss of generality that |V (G)| > 2. By Lemma 5.6,we know that π := ι∼ : G → G′ := G/ι is a non-degenerate harmonic morphism of degree 2.It remains to show that G′ is a tree. Since π ι = π, we have ι∗(π∗(ω′)) = π∗(ω′) for everyω′ ∈ H1(G′) by functoriality. Since ι∗ ≡ −1 on H1(G), we conclude that π∗(ω′) = −π∗(ω′), andtherefore π∗(ω′) = 0, for every ω′ ∈ H1(G′). But π∗ : H1(G′) → H1(G) is injective, so it followsthat H1(G′) = 0, i.e., G′ is a tree.

Remark 5.20. Combining Proposition 4.23 with the proof of (1) ⇒ (5) in Theorem 5.19 yieldsanother proof of Corollary 5.15 (i.e., of the uniqueness of the hyperelliptic involution).

As an application of Theorem 5.19, we establish a special case of [Bak07, Conjecture 3.14]. Tostate the result, given a graph G and a positive integer k, we define σk(G) to be the graph obtainedby replacing each edge of G by a path consisting of k edges.

Corollary 5.21. Let G be a graph, and let k be a positive integer. Then G is hyperelliptic if andonly if σk(G) is hyperelliptic.

Proof. By Corollary 5.11, we may assume without loss of generality that G (and therefore G′ :=σk(G) as well) is a 2-edge-connected graph of genus at least 2. If G is hyperelliptic, then byTheorem 5.19 there is an automorphism ι of G which acts as −1 on H1(G). Identifying V (G) witha subset of V (G′) in the obvious way induces an isomorphism between H1(G) and H1(G′), and it iseasy to see that ι can be extended to an automorphism of G′ which acts as −1 on H1(G′). ThereforeG′ is hyperelliptic. Conversely, suppose that G′ is hyperelliptic. Then there is an automorphismι′ of G′ which acts as −1 on H1(G′). By an argument similar to the proof of Proposition 4.23, itfollows that ι′ induces an automorphism ι of G which acts as −1 on H1(G) (the key point is thatevery cycle in G′ contains a vertex of degree at least 3, which must belong to V (G), and whichmust be sent by ι to another such vertex). Therefore G is hyperelliptic as well.

5.4. The canonical map and 3-edge-connectivity. We now turn to a discussion of a graph-theoretic analogue of the “canonical map” from a Riemann surface to projective space. In algebraicgeometry, the following are equivalent for a Riemann surface X of genus at least 2: (i) X is nothyperelliptic; (ii) the symmetric square S(2) : Div2

+(X) → Jac(X) of the Abel-Jacobi map isinjective; and (iii) the canonical map ψX : X → P(Ω1(X)) is injective. We have already seenthat the analogues of (i) and (ii) are not equivalent for 2-edge-connected graphs of genus at least2; indeed, by Theorem 1.9, S(2) : Div2

+(G) → Jac(G) is injective if and only if G is 3-edge-connected, and this is, by Lemma 5.3, a strictly weaker condition than G being non-hyperelliptic(if |V (G)| > 2). We now define a graph-theoretic version ψG of the canonical map, and show thatthe analogues of conditions (ii) and (iii) for graphs are equivalent. In other words, we will showthat ψG is injective if and only if G is 3-edge-connected.

Let G be a 2-edge-connected graph, and let H1(G) be the space of harmonic 1-forms on G, asdefined in §4. We write P(H1(G)) for the projective space consisting of all hyperplanes (linearsubspaces of codimension 1) in H1(G). We define the canonical map ψG : E(G) → P(H1(G)) bysending an edge e ∈ E(G) to the hyperplane W (e) := ω ∈ H1(G) : ω(e) = 0. Note that


the condition ω(e) = 0 is independent of the orientation of e, so it makes sense to ask whether ornot ω vanishes on an undirected edge. Also, the fact that G is 2-edge-connected guarantees thatW (e) 6= H1(G), so W (e) is indeed a hyperplane.

Our main observation about the canonical map is the following proposition:

Proposition 5.22. Let G be a 2-edge-connected graph. Then the following are equivalent:(1) The canonical map ψG : E(G) → P(H1(G)) is injective.(2) The map S(2) : Div2

+(G) → Jac(G) is injective.(3) G is 3-edge-connected.

Proof. We already know by Theorem 1.9 that (2) ⇔ (3), so it suffices to prove that (1) ⇔ (3).Suppose first that G is 3-edge-connected, and let e1, e2 ∈ E(G). Since G − e1, e2 is connected,there is a cycle C containing e1 but not e2. The characteristic function χC of C is then a flowbelonging to W (e2) but not W (e1), from which it follows that ψG is injective.

Conversely, suppose G is not 3-edge-connected. Then there exist edges e1, e2 ∈ E(G) such thatG−e1, e2 is disconnected. It follows that any flow ω ∈ H1(G) which is non-zero on e1 must alsobe non-zero on e2. Thus W (e1) = W (e2), and ψG is not injective.

Remark 5.23. One can define an analogue ψG,A of the canonical map for flows with values in anarbitrary abelian group A, and certain graph-theoretic assertions about A-flows translate nicely intostatements about ψG,A. For example, for A = Z/5Z, Tutte’s famous 5-flow conjecture (c.f. [Bol98,§X.4, p. 348]) is equivalent to the assertion that if G is a 2-edge-connected graph, then the image ofψG,Z/5Z : E(G) → P(H1(G,Z/5Z)) is contained in an affine subspace (i.e., there exists a hyperplanein P(H1(G,Z/5Z)) disjoint from ψG,Z/5Z(E(G))).

5.5. Hyperelliptic graphs without Weierstrass points. We conclude by using Theorem 5.12and the Riemann-Hurwitz formula for graphs to give a complete characterization of all hyperellipticgraphs having no Weierstrass points. (Graphs with no Weierstrass points are quite interesting fromthe point of view of arithmetic geometry, c.f. [Bak07, Corollary 4.10].)

Recall from [Bak07] that, by analogy with the theory of Riemann surfaces, a vertex x ∈ V (G) iscalled a Weierstrass point if r(g(x)) ≥ 1, where g := g(G). An example is given in [Bak07] of a familyof graphs of genus at least 2 with no Weierstrass points, namely the family Bn = B(1, 1, . . . , 1)consisting of two vertices joined by n ≥ 3 edges. This is in contrast to the classical situation, inwhich every Riemann surface of genus at least 2 has Weierstrass points. (It is also proved in in[Bak07] that every metric graph of genus at least 2 does have Weierstrass points.)

Remark 5.24. On a hyperelliptic Riemann surface X, the Weierstrass points are precisely the fixedpoints of the hyperelliptic involution. For a 2-edge-connected graph G, it is easy to see that a fixedpoint of the hyperelliptic involution is a Weierstrass point, and if g(G) = 2 then the converse alsoholds. However, if g(G) ≥ 3 then the converse does not always hold, as the following exampleshows. Let G be the hyperelliptic graph B(3, 3, 3, 3) of genus 3, and let x, y ∈ V (G) be the internalvertices of one of the edges of G. Then it is not hard to verify that x and y are Weierstrass points.Since ι(x) = y, we see that these points are not fixed by the hyperelliptic involution ι on G.

It turns out that apart from a few exceptions, hyperelliptic graphs almost always have Weierstrasspoints. One exception is the family of graphs Bn mentioned above. Another is the family of graphsB(l1, l2, l3), where l1, l2, l3 are odd positive integers (c.f. Example 5.2). A third exception is thefamily of graphs Φ(l) described in the next paragraph.


For every integer l ≥ 1, let the graph Φ(l) consist of two disjoint paths P = [x0, x1, . . . , xl] andQ = [y0, y2, . . . , yl] of length l, together with two pairs of parallel edges joining x0 to y0 and xl toyl, respectively. It is easy to verify that for the unique involution ι : G → G sending xi to yi, thequotient graph Φ(l)/ι is isomorphic to a path of length l. Thus Φ(l) is hyperelliptic for all l.

Remark 5.25. It follows from Corollary 5.10 that x ∈ V (G) is a Weierstrass point if and only ifρ(x) ∈ V (G) is a Weierstrass point, where G is the 2-edge-connected graph obtained by contractingevery bridge of G. So without loss of generality, when studying Weierstrass points on graphs itsuffices to consider graphs which are 2-edge-connected.

Theorem 5.26. The following are the only 2-edge-connected hyperelliptic graphs with no Weier-strass points:

(1) The graph Bn for some integer n ≥ 3.(2) The graph B(l1, l2, l3) for some odd integers l1, l2, l3 ≥ 1.(3) The graph Φ(l) for some integer l ≥ 1.

Proof. Let G be a 2-edge-connected hyperelliptic graph with no Weierstrass points. If |V (G)| = 2,then G is isomorphic to Bn for some n ≥ 3, so without loss of generality, we may assume that|V (G)| > 2. By Theorem 5.12, there exists a non-degenerate degree 2 harmonic morphism φ : G→T for some tree T with |V (T )| > 1. Moreover, in the proof of Theorem 5.12 we have shown thatfor such φ we have r(φ∗((t))) = 1 for every t ∈ V (T ). If mφ(x) = 2 for some x ∈ V (G), then xis a Weierstrass point, as r(g(x)) ≥ r(2(x)) = r(φ∗(φ(x))) = 1. Therefore, mφ(x) = 1 for everyx ∈ V (G), so that every t ∈ V (T ) has exactly two preimages under φ. By the Riemann-Hurwitzformula for graphs (Theorem 2.14), we have

∑x∈V (G) vφ(x) = 2g + 2.

Consider a vertex t ∈ V (T ) with deg(t) = 1. Let φ−1(t) = x, x′, and let x′′ be the uniqueneighbor of x in V (G) r φ−1(t) (which is well-defined since mφ(x) = 1). It is easy to see thatthere are vφ(x) + 1 edges incident to x, namely the vφ(x) vertical edges connecting x to x′ and thehorizontal edge connecting x to x′′. Also, since G is 2-edge-connected, we have deg(x) ≥ 2, so by(2.2) we know that vφ(x) ≥ 1. It follows that

(vφ(x) + 2)(x) ∼ (x) + vφ(x)(x′) + (x′′) ≥ (x) + (x′) = φ∗(t).

Therefore, as r(φ∗(t)) ≥ 1, we have r((vφ(x) + 2)(x)) ≥ 1, and x is a Weierstrass point of G ifvφ(x) ≤ g − 2. Thus vφ(x) ≥ g − 1 for every x ∈ V (G) such that deg(φ(x)) = 1. Let k := |t ∈V (T ) | deg(t) = 1| ≥ 2. We have 2g + 2 =

∑x∈V (G) vφ(x) ≥ 2k(g − 1) ≥ 4(g − 1). It follows that

either g = 2, or else g = 3 and k = 2. In the latter case, T is a path and vφ(x) = 0 for everyx ∈ V (G) such that deg(φ(x)) > 1.

If g = 2, it is easy to see that G must be isomorphic to the graph B(l1, l2, l3) for some integersl1, l2, l3 ≥ 1, and if li is even for some i ∈ 1, 2, 3 then the middle vertex of the path of length liis a Weierstrass point by Example 5.2. If g = 3, then by the above we have vφ(x) = 2 for everyx ∈ V (G) such that deg(φ(x)) = 1. It follows easily that G is isomorphic to the graph Φ(|E(T )|).

It remains to show that if G is one of the graphs in (1), (2) or (3), then it has no Weierstrasspoints. We start by considering the case when G satisfies (1), i.e., |V (G)| = 2 and |E(G)| = n ≥ 3.Let V (G) = x, y. By Theorem 1.5 we have r((n − 1)(x) − (y)) = −1. Therefore r(g(x)) =r((n − 1)(x)) ≤ r((n − 1)(x) − (y)) + 1 = 0. It follows that x is not a Weierstrass point, and bysymmetry neither is y.

Now suppose that G is of the form (2), so that g = 2. As each li is odd, the hyperellipticinvolution ι on G has no fixed points (since, in the notation of Example 5.2, we have ι(x) = y andι(zi) = zl−i). Therefore G has no Weierstrass points by Remark 5.24.


Finally, suppose that G is of the form (3), i.e., that G is isomorphic to the graph Φ(l) for someinteger l ≥ 1. Let the vertices of G be labeled as in the definition of Φ(l). By symmetry, it sufficesto prove that r(3(xi)) = 0 for every integer i such that 0 ≤ i ≤ l. Suppose first that l ≤ 3i ≤ 2l.Then 3(xi) ∼ (x0) + (xl) + (x3i−l). Consider the following linear order < on V (G):

y0 < y1 < · · · < yl < x0 < x1 < · · · < x3i−l−1 < xl < xl−1 < · · · < x3i−l+1 < x3i−l.

The divisor associated to this order is equal to (x0)+(xl)+(x3i−l)− (y0). It follows that r(3(xi)) =r((x0) + (xl) + (x3i−l)) = 0 in this case. Suppose now that 3i < l. Then we have

3(xi) ∼ (x3i) + 2(x0) ∼ (x3i+1) + 2(y0) ∼ (xl) + (y0) + (yl−3i−1).

This time, we consider the following linear order < on V (G):

yl < xl < xl−1 < · · · < x1 < x0 < y0 < y1 < · · · < yl−3i−2 < yl−1 < yl−2 < · · · < yl−3i < yl−3i−1.

The divisor associated to this order is equal to (xl) + (y0) + (yl−3i−1)− (yl). Again it follows thatr(3(xi)) = 0. The last remaining case, where 3i > 2l, follows by symmetry from the case 3i < l.

References

[Bak07] M. Baker. Specialization of linear systems from curves to graphs. Algebra and Number Theory, 2007. toappear. Available at arXiv:math/0701075, 31 pages.

[BdlHN97] R. Bacher, P. de la Harpe, and T. Nagnibeda. The lattice of integral flows and the lattice of integral cutson a finite graph. Bull. Soc. Math. France, 125(2):167–198, 1997.

[Big97] N. Biggs. Algebraic potential theory on graphs. Bull. London Math. Soc., 29(6):641–682, 1997.[BN07] M. Baker and S. Norine. Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math., 215(2):766–

788, 2007.[Bol98] B. Bollobas. Modern graph theory, volume 184 of Graduate Texts in Mathematics. Springer-Verlag, New

York, 1998.[Che71] W. K. Chen. On vector spaces associated with a graph. SIAM J. Appl. Math., 20:526–529, 1971.[Dha90] D. Dhar. Self-organized critical state of sandpile automaton models. Phys. Rev. Lett., 64(14):1613–1616,

1990.[EF01] J. Eells and B. Fuglede. Harmonic maps between Riemannian polyhedra, volume 142 of Cambridge Tracts

in Mathematics. Cambridge University Press, Cambridge, 2001. With a preface by M. Gromov.[Epp96] D. Eppstein. On the parity of graph spanning tree numbers. Technical report 96-14. Available at

http://www.ics.uci.edu/ eppstein/pubs/Epp-TR-96-14.pdf, 8 pages, 1996.[Lor91] D. J. Lorenzini. A finite group attached to the Laplacian of a graph. Discrete Math., 91(3):277–282, 1991.[Mir95] R. Miranda. Algebraic curves and Riemann surfaces, volume 5 of Graduate Studies in Mathematics.

American Mathematical Society, Providence, RI, 1995.[Ura00] H. Urakawa. A discrete analogue of the harmonic morphism and Green kernel comparison theorems.

Glasg. Math. J., 42(3):319–334, 2000.

School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160, USAE-mail address: [email protected]

Department of Mathematics, Princeton University, Princeton, New Jersey, 08544-1000, USAE-mail address: [email protected]

Documents

Introduction - Georgia Institute of Technologypeople.math.gatech.edu/~mbaker/pdf/hyperelliptic.pdf[Bak07].) We then prove that the graph-theoretic analogues of conditions (H1)-(H5)